Microsoft Word - numero_59_art_03_3255.docx A. Behtani et alii, Frattura ed Integrità Strutturale, 59 (2022) 35-48; DOI: 10.3221/IGF-ESIS.59.03 35 Focussed on Steels and Composites for Engineering Structures Residual Force Method for damage identification in a laminated composite plate with different boundary conditions Amar Behtani, Samir Tiachacht Laboratory of Mechanics, Structure, and Energetics, Mouloud Mammeri University of Tizi-Ouzou, Algeria. amar.behtani@ummto.dz; https://orcid.org/0000-0003-4503-0409 samir.tiachacht@ummto.dz; https://orcid.org/0000-0002-3597-8725 Tawfiq Khatir Department of Technology, University Centre of Naama - Salhi Ahmed, Naama, Algeria Khatir_usto@yahoo.com; https://orcid.org/0000-0003-3009-8371 Samir Khatir, Magd Abdel Wahab Soete Laboratory, Faculty of Engineering and Architecture, Ghent University, Technologiepark Zwijnaarde, Zwijnaarde, Belgium. Khatir.samir@hotmail.fr; https://orcid.org/0000-0002-8101-3633 Magd.AbdelWahab@UGent.be; https://orcid.org/0000-0002-3610-865X Brahim Benaissa Design Engineering Laboratory, Toyota Technological Institute, Nagoya, Japan. Benaissa@toyota-ti.ac.jp; https://orcid.org/0000-0002-9472-9331 Mohand Slimani Laboratory of Mechanics, Structure, and Energetics, Mouloud Mammeri University of Tizi-Ouzou, Algeria. mohand.slimani@ummto.dz; https://orcid.org/0000-0003-4252-4892 ABSTRACT. The strongest point about damage identification based on the dynamic measurements, is the ability to perform structural health evaluation globally. Researchers in the last few years payed more attention to damage indicators based on modal analysis using either frequencies, mode shapes, or Frequency Response Functions (FRFs). This paper presents a new application of damage identification in a cross-ply (0°/90°/0°) laminated composite plate based on Force Residual Method (FRM) damage indicator. Considering single and multiple damages with different damage levels. As well as investigating the SSSS and CCCC boundary conditions effect on the estimation accuracy. Moreover, a white Gaussian noise is introduced to test the challenges of the technique. The results show that the suggested FRM indicator provides accurate results under different boundary conditions, favouring the SSSS boundary condition than the CCCC for 3% noise. Citation: Behtani, A., Tiachacht, S., Khatir, T., Hhatir, S., Abdel Wahab, M., Benaissa, B., Slimani, M., Residual Force Method for damage identification in a laminated composite plate with different boundary conditions, Frattura ed Integrità Strutturale, 59 (2022) 35-48. Received: 28.08.2021 Accepted: 02.10.2021 Published: 01.01.2022 Copyright: © 2022 This is an open access article under the terms of the CC-BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. https://youtu.be/23QmYy-OzD8 A. Behtani et alii, Frattura ed Integrità Strutturale, 59 (2022) 35-48; DOI: 10.3221/IGF-ESIS.59.03 36 KEYWORDS. Damage quantification; RFM; Noise; Composite laminated plate. INTRODUCTION omposite materials become a cornerstone of modern material and are being used in almost every engineering discipline, like civil, infrastructure, aerospace engineering to name a few, due to their outstanding strength compared to their weight. However, composite materials are not immune to damage, and when it happens, whether due to fatigue or accidents, it can reduce their rigidity significantly. Therefore, researchers in the last decade developed new methods for damage identification dedicated to composite materials. In such methods, vibration-based structural responses are commonly used. Doebling et al. [1] presented a review of these approaches. We mention in this paper some of the noticeable literature, such as the study made by Khatri et al [2], in which the authors considered experimental analysis for damage identification in a complex structure, with an inverse problem formulation based on Particle Swarm Optimization (PSO). Ghannadi et al. [3] presented an approach based on the Multiverse Optimizer (MVO). Where two objective functions were used, namely the modal assurance criterion (MAC) and the modified total modal assurance criterion (MTMAC). Wand et al. [4] Suggested a neural network technique for damage identification refinement in the case of a suspension bridge. And Providakis et al. [5] used impedance-type measurements and error statistics to present a new damage identification approach in composite structures. Vahedian et al. [6] considered the case of multi-storey timber buildings and presented an improvement on the damage assessment by SFRP. And Tiachacht et al. [7] investigated a new modified indicator based on Cornwell Indicator (CI). This indicator was tested in different structures and the results suggested better accuracy than the peer methods. The frequency change-based approach was used in the classification of vibration-based damage detection techniques [8, 9], a technique based on curvature mode shape [10], as well as another technique based on Modal Strain Energy [11]. Computational cost is often an issue in such methods. A quick damage identification method in the laminated composite plate using CI and Machine learning based Artificial Neural Network (ANN) was presented in Ref. [12]. The authors used Isogeometric Analysis (IGA) combined with damage indicator for damage localization and ANN for damage quantification. Different indicators were combined with laminated composite beams and plates in Refs. [13, 14]. Nobahari et al. [15] suggested an approach with the name: “Flexibility Strain Energy-Based Index (FSEBI)”. For multiple damages identification, in both simple and complex cases. The efficiency of this method for multiple damage localization was shown in the results. Flexibility matrix for damage identification was investigated by [16]. This technique is efficient for damaged detection. However, the provided technique is experimentally validated. Ghannadi et al. [17] considered a variety of structures to investigate their new approach based on the Grey Wolf Optimization (GWO) algorithm. Using an objective function that combines the frequencies and mode shapes. The validation of the suggested methods is carried out in two experimental studies, namely a cantilever beam and a truss tower. Different optimization techniques were investigated for damage identification problems [2, 12, 18-20], such as steel and composite plates, steel and composite beams, and complex structures. In this paper, a laminated composite plate with three layers (0°/90°/0°) is considered for damage identification, using the residual force method with FEM. The structural response is studied under two kinds of boundary conditions, namely, fully simply supported (SSSS) and fully clamped plates (CCCC) to test the accuracy of FRM. Considering the measurement uncertainty up to 3% level, simulated by white Gaussian noise. METHODOLOGY Finite Element Analysis of laminate plate his study is based on the First-order shear deformation theory (FSDT) [21]. In such a method, after material deformation, the effects of transverse shear strains cause the transverse to not remain perpendicular to the mid surface. w is not function of the thickness coordinate z because of the inextensibility of transverse normal. Therefore, the displacement field of the FSDT function of time t is written in the following form: C T A. Behtani et alii, Frattura ed Integrità Strutturale, 59 (2022) 35-48; DOI: 10.3221/IGF-ESIS.59.03 37                           0 0 0 , , , , , , , , , , , , , , , , , , , x y u x y z t u x y t z x y t v x y z t v x y t z x y t w x y z t w x y t (1) where  0 0 0, ,u v w is the displacement vector of a point on the plane  0z , x is the rotations of a transverse normal around the x-axes and  y is the equivalent for the y-axes. Figure 1: Laminated plate - organization of layers in the thickness direction. The strain energy is:             21 2 1 2 T T k T k T k T k m m m f f m f f A z T T k c c c A z U u B D B B D B B zD B B z D B dzudA u B D B dzudA (2) The components of the stiffness matrix include the membrane-bending coupling part, are written as follows:                e e e e e emm mf fm ff ccK K K K K K (3) where emmK is the stiffness matrix of the membrane part, and e mfK , e fmK are the membrane-bending coupling components, e ffK represent the bending component and e ccK represent the shear component. These matrices are detailed as follows:                                                       1 1 2 2 1 1 2 2 1 1 3 3 1 1 1 1 1 2 1 2 1 3 c c c c c n e T mm m k m k k k A n e T mf m k f k k k A n e T fm f k m k k k A n e T ff f k f k k k A n e T cc c c c k k k A K B D B z z dA K B D B z z dA K B D B z z dA K B D B z z dA K B D B z z dA (4) Here cn represents the number of layers across the z-axes. z Layer 1 Layer k zk+1 zk A. Behtani et alii, Frattura ed Integrità Strutturale, 59 (2022) 35-48; DOI: 10.3221/IGF-ESIS.59.03 38 There exist three B matrices (membrane, the bending and the shear components), corresponding to the strain-displacements. They are described in detail as follows:                                     , , , , , , , , , , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x m y y x x f y y x x c y N B N N N N B N N N N N B N N (5) where   , x N N x and   , y N N y In this paper, we consider the dynamic version of the principle of virtual displacements. The equations of motion for the free vibration of symmetric cross-ply laminated plates can be expressed as:                                                                      2 2 2 55 44 02 2 2 222 22 11 12 66 55 22 2 2 2 22 2 12 22 66 442 2 2 x x s y x s x y yx x s w w w K A A I x xx y t yw wxD D D K A I x y x y xx y t w D D D K A x y yy y x                    2 2 2 y y I t (6) Here, 𝐼 and 𝐼 are the tensor components for the mass inertia, and they are defined as:    3 0 2; 12 h I h I (7) where  and h are, respectively, the density and the total thickness of the composite plate. 𝐴 and 𝐷 are the extensional and bending stiffnesses, and they are expressed as in the following equations:                     1 1 3 3 1 1 N k ij ij k k k N k ij ij k k k A Q z z D Q z z (8) where  kijQ is the stress-reduced stiffness matrix for the transformed material plane in each layer k. In Eqn. (8), the matrix can be obtained as follows: A. Behtani et alii, Frattura ed Integrità Strutturale, 59 (2022) 35-48; DOI: 10.3221/IGF-ESIS.59.03 39  TmQ PQ P (9) where P is the transformation matrix given by:                             2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 P (10) where    cos ;    sin and   2 sin 2 and mQ is the stress-reduced stiffness for the material plane                      1 12 2 12 2 2 23 13 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 m GQ G G (11) Here        1 1 12 211 E ; and        2 2 12 211 E , where 1E is Young’s modulus for a layer parallel to fibres. And 2E is the equivalent Young’s modulus for the layers that are perpendicular to fibres. On the other hand, 12 and 21 are Poisson’s ratios. 23G is the shear modulus for planes 2−3, 13G is the equivalent shear modulus for planes 1−3, and 12G represent the shear modulus for planes 1−2. The variables w , x and  y can be expressed in the harmonic forms as follows:                           , , , , , , , , , i t i t x x i t y y w x y t x y e x y t x y e x y t x y e (12) and the equations of motion (6) become:                                                                                  2 2 2 55 44 02 2 2 222 2 11 12 66 55 22 2 2 22 2 12 22 66 442 2 2 x x s y yx s x x y yx x s y K A A I x xx y D D D K A I x y x y xx y D D D K A x y yy y x                2 2 yI (13) In which  is natural frequency. A. Behtani et alii, Frattura ed Integrità Strutturale, 59 (2022) 35-48; DOI: 10.3221/IGF-ESIS.59.03 40 Residual forces In this section, we explore structural damage identification based on the residual forces, details about residual forces can be found in Ref. [22]. In such approaches, the structure is divided into small elements via the finite element method, and the damage index of each element is expressed as the change of the rigidity i.e.:                  e e ee u d uii i i iK K K K (14) where    eu i K and    ed i K are the elementary matrices of the undamaged and damaged structures, respectively. In the above equation, Eqn. (14),   eiK is the difference in stiffness, and  is a value between 0 and 1, which indicates a loss of rigidity in each element. In other words, for undamaged elements,  is equal to 0, and to 1 in fully damaged elements. In this study, we assume that the damage does not affect the mass matrix of a damaged structure, therefore the rigidity matrix of the damaged element is expressed as follows:                  0 1, , ee u jj j M K K j m (15) and the modal residual force vector is expressed by this equation:                                                  1 2 1 2 e e ed i ii mi m R K f f f f F (16) As consequence, Eqn. (16) can be represented in a matrix form by the following expression:      F R (17) The coefficient matrix  F is:        eeu d ij j ij F K (18) Here,   ij F is the force vector in the actual node thi on the thj element, written in the global coordinates. The vector of residual force modal is expressed in this manner:           d dii iR K M (19) and Eqn. (17) is rewritten in this manner:                                                                    111 12 1 1 221 22 2 2 1 2 m m mn n nm n F F F R F F F R F F F R (20) A. Behtani et alii, Frattura ed Integrità Strutturale, 59 (2022) 35-48; DOI: 10.3221/IGF-ESIS.59.03 41 Here, n is the number of modes. And m is the number of elements. And the damage indicators are computed through the system of equations Eqn. (20)                                                                     1 11 12 1 1 2 21 22 2 2 1 2 m m m n n nm n F F F R F F F R F F F R (21) NUMERICAL EXAMPLES A square laminated (0°/90°/0°) plates o investigate the performance of the suggested method, we consider an example of square cross-ply laminated plates. In this study, we assume that all layers of the laminate have the same thickness and density, and made of linearly elastic composite material, with the following properties for each layer:   1 2 12 13 2/ 40, 0.6E E G G E ; 23 2 0.5G E ;  12 0.25 . Subscripts 1 denote the direction parallel to fibre orientation in a layer and 2 represent the perpendicular direction to the fibre. Considering that the measurement is performed clockwise, the ply angle is positive of each layer from the global x-axis to the fibre direction, and negative if measured anti-clockwise. we consider a mesh of 10 ×10 (100 elements) as shown in Fig. 2. And to obtain the natural frequencies and mode shapes, the eigenproblem is solved using MATLAB. We compare our results to those published by Ferreira and Fasshauer [23] and [24], considering the same shear correction factors, as well as the nondimensionalized natural frequencies. Figure 2: (a) A cross-ply (0°/90°/0°) square composite plate and (b) Discretized square isotropic plate with two damage scenarios. The Shear correction factor is considered as  2 / 12sK , and the dimensionless natural frequency is given by:      2 2 0b h D where    30 2 12 2112 1D E h . We also consider in this study, two damage scenarios, simulated by reducing the global stiffness of individual element in the following manner:      1 1 nele d e e K K (22) Here, represents the damage ratio. dK is the global stiffness matrix representing the damaged structures. Finally, eK is the stiffness matrix of the eth element. z x y Lx Ly 0° 90° 0° (a) 1 2 10 91 92 100 Scenario 2 32 2928 8 Lx= 1 m L y= 1 m Scenario 1 x y Scenario 4Scenario 3 (b) 16 24 40 63 64 T A. Behtani et alii, Frattura ed Integrità Strutturale, 59 (2022) 35-48; DOI: 10.3221/IGF-ESIS.59.03 42 Considering three-ply (0°/90°/0°) clamped CCCC square, with laminated plates of thickness to span ratios  0.2t b and  1a b , Tab. 2 shows the convergence results of dimensionless natural frequencies. To validate the proposed damage identification approach, we consider four damage scenarios. The first scenario has one damage, and two damages in the second scenario and two other scenarios has three and four damages, respectively. Details of each damage case are shown in Tab. 1. And the corresponding natural frequencies are shown in Tab. 2. Fig. 3 shows the first four mode shapes of the cross-ply (0°/90°/0°) square composite plate under SSSS boundary conditions. Scenario 1 Scenario 2 Scenario 3 Scenario 4 Element No 16 28 29 8 16 24 32 40 63 64 Severity % 15 5 5 15 5 10 10 5 5 10 Table 1: Damage scenarios. Mode 1 2 3 4 5 Intact [23] 10 × 10 3.5479 5.8947 7.3163 8.6545 9.7538 [24] 3.5939 5.7691 7.3972 8.6876 9.1451 Present 10 × 10 3.5479 5.8947 7.3163 8.6545 9.7538 Damaged Scenario 1 3.5425 5.8934 7.3093 8.6516 9.7514 Scenario 2 3.5470 5.8933 7.3162 8.6541 9.7518 Scenario 3 3.5410 5.8885 7.3048 8.6377 9.7461 Scenario 4 3.5464 5.8905 7.3102 8.6455 9.7392 Table 2: Naturel frequencies of undamaged and damaged laminate plate structure SSSS. Figure 3: The first four mode shapes of the cross-ply (0°/90°/0°) fully simply supported square composite plate SSSS. A. Behtani et alii, Frattura ed Integrità Strutturale, 59 (2022) 35-48; DOI: 10.3221/IGF-ESIS.59.03 43 (a) Noise-free. (b) Noise 3%. Figure 4: Damage index for all elements of the cross-ply (0°/90°/0°) square composite plate SSSS – Scenario 1. (a) Noise-free. (b) Noise 3%. Figure 5: Damage index for all elements of the cross-ply (0°/90°/0°) square composite plate SSSS – Scenario 2. (a) Noise-free. (b) Noise 3%. Figure 6: Damage index for all elements of the cross-ply (0°/90°/0°) square composite plate SSSS – Scenario 3. A. Behtani et alii, Frattura ed Integrità Strutturale, 59 (2022) 35-48; DOI: 10.3221/IGF-ESIS.59.03 44 (a) Noise-free. (b) Noise 3%. Figure 7: Damage index for all elements of the cross-ply (0°/90°/0°) square composite plate SSSS – Scenario 4. The presented results showed that FRM predicts the location of damage correctly with higher accuracy in a laminated composite plate in the case of SSSS boundary conditions. On the other hand, the indicator can support the noise of up to 3%. We note that prediction error under 3% noise correlates with the number of damages. The prediction error in the case of single and double damaged plates is very small, which is noticed when the number of damages is higher than three. Fig. 8 shows the first four mode shapes, in the case of cross-ply (0°/90°/0°) square composite plate under CCCC boundary conditions. And Figs. 9-12 plot the results of each scenario in the presence and absence of measurement noise. The natural frequencies of each damage scenario are presented in Tab. 3. Mode 1 2 3 4 5 Intact [23] 10 × 10 4.4145 6.8373 7.6291 9.2078 10.3964 [24] 4.4468 6.6419 7.6996 9.1852 9.7378 Present 10 × 10 4.4145 6.8373 7.6291 9.2078 10.3964 Damaged Scenario 1 4.4069 6.8359 7.6198 9.2046 10.3939 Scenario 2 4.4132 6.8347 7.6285 9.2060 10.3941 Scenario 3 4.4060 6.8299 7.6169 9.1899 10.3885 Scenario 4 4.4109 6.8282 7.6238 9.1976 10.3775 Table 3: Naturel frequencies of undamaged and damaged laminated plate structure CCCC. Under boundary condition CCCC, the indicator can support the noise of up to 3%. However, the prediction error under noise is different in each scenario. The prediction error in the case of single and double damaged plates is higher than that of the cases of SSSS. Fig. 13 summarizes the obtained damage identification results for the laminated composite plate with and without noise and the accuracy in each damage scenario. These results suggest that the scenario of plates under SSSS boundary condition correspond to the same accuracy in all cases where there is no noise. However, it has an advantage over CCCC in the scenarios of 3% noise and three damages (Case 3). CCCC on the other hand presents an advantage in the small number of damages. A. Behtani et alii, Frattura ed Integrità Strutturale, 59 (2022) 35-48; DOI: 10.3221/IGF-ESIS.59.03 45 Figure 8: The first four mode shapes of the cross-ply (0°/90°/0°) fully clamped square composite plate CCCC. (a) Noise-free. (b) Noise 3%. Figure 9: Damage index for all elements of the cross-ply (0°/90°/0°) square composite plate CCCC – Scenario 1. A. Behtani et alii, Frattura ed Integrità Strutturale, 59 (2022) 35-48; DOI: 10.3221/IGF-ESIS.59.03 46 (a) Noise-free. (b) Noise 3%. Figure 10: Damage index for all elements of the cross-ply (0°/90°/0°) square composite plate CCCC – Scenario 2. (a) Noise-free. (b) Noise 3%. Figure 11: Damage index for all elements of the cross-ply (0°/90°/0°) square composite plate (CCCC) – Scenario 3. (a) Noise-free. (b) Noise 3%. Figure 12: Damage index for all elements of the cross-ply (0°/90°/0°) square composite plate (CCCC) – Scenario 4. A. Behtani et alii, Frattura ed Integrità Strutturale, 59 (2022) 35-48; DOI: 10.3221/IGF-ESIS.59.03 47 Figure 13: Comparison of obtained damage identification results for the laminated composite plat without noise and with noise. CONCLUSION n this study, we present an application for damage identification based on Force Residual Method (FRM) in the case of a cross-ply (0°/90°/0°) laminated composite plate. We consider the cases of single and multiple damages, and different levels of damages. To test the accuracy of FRM, different boundary conditions were applied to the composite plate. 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